Mixing
$limsup_{ntoinfty}{a_n}^{b_n}= limsup_{ntoinfty}{a_n}^{limlimits_{ntoinfty}b_n}$ under certian conditions.
$begingroup$
I have a proof that requires the following justification. So, I decided to recast it in the following form:
If $limlimits_{ntoinfty}b_n$ exists and $limsup_{ntoinfty}{a_n}=a>0$, then $limsup_{ntoinfty}{a_n}^{b_n}= limsup_{ntoinfty}{a_n}^{limlimits_{ntoinfty}b_n}$
My proof
begin{align} limsup_{ntoinfty}{a_n}^{b_n}&=limsup_{ntoinfty}e^{ln{a_n}^{b_n}}\&=limsup_{ntoinfty}e^{b_nln{a_n}}\&=e^{limsup_{ntoinfty}(b_nln{a_n})}\&=e^{limlimits_{ntoinfty}b_nlimsup_{ntoinfty}ln{a_n}}\&=e^{limlimits_{ntoinfty}b_nln{(limsup_{ntoinfty}a_n)}}\&=e^{limlimits_{ntoinfty}b_nln{a}}\&=e^{ln{a}^{limlimits_{ntoinfty}b_n}}\&={a}^{limlimits_{ntoinfty}b_n}\&={limsup_{ntoinfty}{a_n}}^{limlimits_{ntoinfty}b_n}end{align}
Please, is this correct? And is there any other proof?
real-analysis limsup-and-liminf
asked Jan 30 at 9:38
MichealMicheal
26511
$endgroup$
add a comment |
$begingroup$
I have a proof that requires the following justification. So, I decided to recast it in the following form:
If $limlimits_{ntoinfty}b_n$ exists and $limsup_{ntoinfty}{a_n}=a>0$, then $limsup_{ntoinfty}{a_n}^{b_n}= limsup_{ntoinfty}{a_n}^{limlimits_{ntoinfty}b_n}$
My proof
begin{align} limsup_{ntoinfty}{a_n}^{b_n}&=limsup_{ntoinfty}e^{ln{a_n}^{b_n}}\&=limsup_{ntoinfty}e^{b_nln{a_n}}\&=e^{limsup_{ntoinfty}(b_nln{a_n})}\&=e^{limlimits_{ntoinfty}b_nlimsup_{ntoinfty}ln{a_n}}\&=e^{limlimits_{ntoinfty}b_nln{(limsup_{ntoinfty}a_n)}}\&=e^{limlimits_{ntoinfty}b_nln{a}}\&=e^{ln{a}^{limlimits_{ntoinfty}b_n}}\&={a}^{limlimits_{ntoinfty}b_n}\&={limsup_{ntoinfty}{a_n}}^{limlimits_{ntoinfty}b_n}end{align}
Please, is this correct? And is there any other proof?
real-analysis limsup-and-liminf
asked Jan 30 at 9:38
MichealMicheal
26511
$endgroup$
$begingroup$
Proofs shouldn't just be equations, they should be sentences. For example, a sentence about why line $2$ and line $3$ are equal...
$endgroup$
– 5xum
Jan 30 at 9:39
$begingroup$
@5xum: Sorry, it was a typo.
$endgroup$
– Micheal
Jan 30 at 9:45
$begingroup$
@5xum: I edited it.
$endgroup$
– Micheal
Jan 30 at 11:37
add a comment |
$begingroup$
I have a proof that requires the following justification. So, I decided to recast it in the following form:
If $limlimits_{ntoinfty}b_n$ exists and $limsup_{ntoinfty}{a_n}=a>0$, then $limsup_{ntoinfty}{a_n}^{b_n}= limsup_{ntoinfty}{a_n}^{limlimits_{ntoinfty}b_n}$
My proof
begin{align} limsup_{ntoinfty}{a_n}^{b_n}&=limsup_{ntoinfty}e^{ln{a_n}^{b_n}}\&=limsup_{ntoinfty}e^{b_nln{a_n}}\&=e^{limsup_{ntoinfty}(b_nln{a_n})}\&=e^{limlimits_{ntoinfty}b_nlimsup_{ntoinfty}ln{a_n}}\&=e^{limlimits_{ntoinfty}b_nln{(limsup_{ntoinfty}a_n)}}\&=e^{limlimits_{ntoinfty}b_nln{a}}\&=e^{ln{a}^{limlimits_{ntoinfty}b_n}}\&={a}^{limlimits_{ntoinfty}b_n}\&={limsup_{ntoinfty}{a_n}}^{limlimits_{ntoinfty}b_n}end{align}
Please, is this correct? And is there any other proof?
real-analysis limsup-and-liminf
asked Jan 30 at 9:38
MichealMicheal
26511
$endgroup$
I have a proof that requires the following justification. So, I decided to recast it in the following form:
If $limlimits_{ntoinfty}b_n$ exists and $limsup_{ntoinfty}{a_n}=a>0$, then $limsup_{ntoinfty}{a_n}^{b_n}= limsup_{ntoinfty}{a_n}^{limlimits_{ntoinfty}b_n}$
My proof
begin{align} limsup_{ntoinfty}{a_n}^{b_n}&=limsup_{ntoinfty}e^{ln{a_n}^{b_n}}\&=limsup_{ntoinfty}e^{b_nln{a_n}}\&=e^{limsup_{ntoinfty}(b_nln{a_n})}\&=e^{limlimits_{ntoinfty}b_nlimsup_{ntoinfty}ln{a_n}}\&=e^{limlimits_{ntoinfty}b_nln{(limsup_{ntoinfty}a_n)}}\&=e^{limlimits_{ntoinfty}b_nln{a}}\&=e^{ln{a}^{limlimits_{ntoinfty}b_n}}\&={a}^{limlimits_{ntoinfty}b_n}\&={limsup_{ntoinfty}{a_n}}^{limlimits_{ntoinfty}b_n}end{align}
Please, is this correct? And is there any other proof?
real-analysis limsup-and-liminf
real-analysis limsup-and-liminf
asked Jan 30 at 9:38
MichealMicheal
26511
asked Jan 30 at 9:38
MichealMicheal
26511
edited Jan 30 at 9:51
Micheal
asked Jan 30 at 9:38
MichealMicheal
26511
asked Jan 30 at 9:38
MichealMicheal
26511
asked Jan 30 at 9:38
MichealMicheal
26511
26511
$begingroup$
Proofs shouldn't just be equations, they should be sentences. For example, a sentence about why line $2$ and line $3$ are equal...
$endgroup$
– 5xum
Jan 30 at 9:39
$begingroup$
@5xum: Sorry, it was a typo.
$endgroup$
– Micheal
Jan 30 at 9:45
$begingroup$
@5xum: I edited it.
$endgroup$
– Micheal
Jan 30 at 11:37
add a comment |
$begingroup$
Proofs shouldn't just be equations, they should be sentences. For example, a sentence about why line $2$ and line $3$ are equal...
$endgroup$
– 5xum
Jan 30 at 9:39
$begingroup$
@5xum: Sorry, it was a typo.
$endgroup$
– Micheal
Jan 30 at 9:45
$begingroup$
@5xum: I edited it.
$endgroup$
– Micheal
Jan 30 at 11:37
$begingroup$
Proofs shouldn't just be equations, they should be sentences. For example, a sentence about why line $2$ and line $3$ are equal...
$endgroup$
– 5xum
Jan 30 at 9:39
$begingroup$
Proofs shouldn't just be equations, they should be sentences. For example, a sentence about why line $2$ and line $3$ are equal...
$endgroup$
– 5xum
Jan 30 at 9:39
$begingroup$
@5xum: Sorry, it was a typo.
$endgroup$
– Micheal
Jan 30 at 9:45
$begingroup$
@5xum: Sorry, it was a typo.
$endgroup$
– Micheal
Jan 30 at 9:45
$begingroup$
@5xum: I edited it.
$endgroup$
– Micheal
Jan 30 at 11:37
$begingroup$
@5xum: I edited it.
$endgroup$
– Micheal
Jan 30 at 11:37
add a comment |
2 Answers
2
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$begingroup$
It's correct because $f(x)=e^x$ and $ln$ are continuous functions.
For all continuous in $a$ function $f$ if $limlimits_{nrightarrow+infty}a_n=a$ then:
$$lim_{nrightarrow+infty}f(a_n)=fleft(lim_{nrightarrow+infty}a_nright)=f(a).$$
answered Jan 30 at 9:49
Michael RozenbergMichael Rozenberg
109k1896201
$endgroup$
add a comment |
$begingroup$
Your proof seems a bit more complicated than it should be. Since $a^b$ is continuous for all $a,bin mathbb{R}$, we have that $limsup_{nrightarrowinfty}a_n^{b_n} = limsup_{nrightarrowinfty}a_n^{limsup_{nrightarrowinfty} b_n} $. However, since $lim_{nrightarrowinfty} b_n$ exists, we have $limsup_{nrightarrowinfty}b_n=lim_{nrightarrowinfty}b_n$, from which your claim follows.
edited Jan 30 at 9:57
Mark
10.4k1622
answered Jan 30 at 9:55
UnexpectedExpectationUnexpectedExpectation
739
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
active
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$begingroup$
It's correct because $f(x)=e^x$ and $ln$ are continuous functions.
For all continuous in $a$ function $f$ if $limlimits_{nrightarrow+infty}a_n=a$ then:
$$lim_{nrightarrow+infty}f(a_n)=fleft(lim_{nrightarrow+infty}a_nright)=f(a).$$
answered Jan 30 at 9:49
Michael RozenbergMichael Rozenberg
109k1896201
$endgroup$
add a comment |
$begingroup$
It's correct because $f(x)=e^x$ and $ln$ are continuous functions.
For all continuous in $a$ function $f$ if $limlimits_{nrightarrow+infty}a_n=a$ then:
$$lim_{nrightarrow+infty}f(a_n)=fleft(lim_{nrightarrow+infty}a_nright)=f(a).$$
answered Jan 30 at 9:49
Michael RozenbergMichael Rozenberg
109k1896201
$endgroup$
add a comment |
$begingroup$
It's correct because $f(x)=e^x$ and $ln$ are continuous functions.
For all continuous in $a$ function $f$ if $limlimits_{nrightarrow+infty}a_n=a$ then:
$$lim_{nrightarrow+infty}f(a_n)=fleft(lim_{nrightarrow+infty}a_nright)=f(a).$$
answered Jan 30 at 9:49
Michael RozenbergMichael Rozenberg
109k1896201
$endgroup$
It's correct because $f(x)=e^x$ and $ln$ are continuous functions.
For all continuous in $a$ function $f$ if $limlimits_{nrightarrow+infty}a_n=a$ then:
$$lim_{nrightarrow+infty}f(a_n)=fleft(lim_{nrightarrow+infty}a_nright)=f(a).$$
answered Jan 30 at 9:49
Michael RozenbergMichael Rozenberg
109k1896201
edited Jan 30 at 9:57
answered Jan 30 at 9:49
Michael RozenbergMichael Rozenberg
109k1896201
answered Jan 30 at 9:49
Michael RozenbergMichael Rozenberg
109k1896201
answered Jan 30 at 9:49
Michael RozenbergMichael Rozenberg
109k1896201
109k1896201
add a comment |
add a comment |
$begingroup$
Your proof seems a bit more complicated than it should be. Since $a^b$ is continuous for all $a,bin mathbb{R}$, we have that $limsup_{nrightarrowinfty}a_n^{b_n} = limsup_{nrightarrowinfty}a_n^{limsup_{nrightarrowinfty} b_n} $. However, since $lim_{nrightarrowinfty} b_n$ exists, we have $limsup_{nrightarrowinfty}b_n=lim_{nrightarrowinfty}b_n$, from which your claim follows.
edited Jan 30 at 9:57
Mark
10.4k1622
answered Jan 30 at 9:55
UnexpectedExpectationUnexpectedExpectation
739
$endgroup$
add a comment |
$begingroup$
Your proof seems a bit more complicated than it should be. Since $a^b$ is continuous for all $a,bin mathbb{R}$, we have that $limsup_{nrightarrowinfty}a_n^{b_n} = limsup_{nrightarrowinfty}a_n^{limsup_{nrightarrowinfty} b_n} $. However, since $lim_{nrightarrowinfty} b_n$ exists, we have $limsup_{nrightarrowinfty}b_n=lim_{nrightarrowinfty}b_n$, from which your claim follows.
edited Jan 30 at 9:57
Mark
10.4k1622
answered Jan 30 at 9:55
UnexpectedExpectationUnexpectedExpectation
739
$endgroup$
add a comment |
$begingroup$
Your proof seems a bit more complicated than it should be. Since $a^b$ is continuous for all $a,bin mathbb{R}$, we have that $limsup_{nrightarrowinfty}a_n^{b_n} = limsup_{nrightarrowinfty}a_n^{limsup_{nrightarrowinfty} b_n} $. However, since $lim_{nrightarrowinfty} b_n$ exists, we have $limsup_{nrightarrowinfty}b_n=lim_{nrightarrowinfty}b_n$, from which your claim follows.
edited Jan 30 at 9:57
Mark
10.4k1622
answered Jan 30 at 9:55
UnexpectedExpectationUnexpectedExpectation
739
$endgroup$
Your proof seems a bit more complicated than it should be. Since $a^b$ is continuous for all $a,bin mathbb{R}$, we have that $limsup_{nrightarrowinfty}a_n^{b_n} = limsup_{nrightarrowinfty}a_n^{limsup_{nrightarrowinfty} b_n} $. However, since $lim_{nrightarrowinfty} b_n$ exists, we have $limsup_{nrightarrowinfty}b_n=lim_{nrightarrowinfty}b_n$, from which your claim follows.
edited Jan 30 at 9:57
Mark
10.4k1622
answered Jan 30 at 9:55
UnexpectedExpectationUnexpectedExpectation
739
edited Jan 30 at 9:57
Mark
10.4k1622
edited Jan 30 at 9:57
Mark
10.4k1622
edited Jan 30 at 9:57
Mark
10.4k1622
10.4k1622
answered Jan 30 at 9:55
UnexpectedExpectationUnexpectedExpectation
739
answered Jan 30 at 9:55
UnexpectedExpectationUnexpectedExpectation
739
answered Jan 30 at 9:55
UnexpectedExpectationUnexpectedExpectation
739
739
add a comment |
add a comment |
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$begingroup$
Proofs shouldn't just be equations, they should be sentences. For example, a sentence about why line $2$ and line $3$ are equal...
$endgroup$
– 5xum
Jan 30 at 9:39
$begingroup$
@5xum: Sorry, it was a typo.
$endgroup$
– Micheal
Jan 30 at 9:45
$begingroup$
@5xum: I edited it.
$endgroup$
– Micheal
Jan 30 at 11:37